Parameter estimation from measurements along quantum trajectories

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Parameter estimation from measurements along quantum trajectories

P. Six, Philippe Campagne-Ibarcq, L Bretheau, Benjamin Huard, Pierre Rouchon

To cite this version:

P. Six, Philippe Campagne-Ibarcq, L Bretheau, Benjamin Huard, Pierre Rouchon. Parameter estima-

tion from measurements along quantum trajectories. 54th IEEE Conference on Decision and Control

(CDC 2015), Dec 2015, Osaka, Japan. �hal-01245085�

arXiv:1503.06149v1 [math.OC] 20 Mar 2015

Parameter estimation from measurements along quantum trajectories

P. Six

Ph. Campagne-Ibarcq

L. Bretheau

B. Huard

P. Rouchon

Abstract

The dynamics of many open quantum systems are described by stochastic master equations. In the discrete-time case, we recall the structure of the derived quantum filter governing the evolution of the density op- erator conditioned to the measurement outcomes. We then describe the structure of the corresponding par- ticle quantum filters for estimating constant parameter and we prove their stability. In the continuous-time (dif- fusive) case, we propose a new formulation of these particle quantum filters. The interest of this new for- mulation is first to prove stability, and also to provide an efficient algorithm preserving, for any discretization step-size, positivity of the quantum states and parameter classical probabilities. This algorithm is tested on ex- perimental data to estimate the detection efficiency for a superconducting qubit whose fluorescence field is mea- sured using a heterodyne detector.

1. Introduction

Parameter estimation in hidden Markov models is a well established subject (see, e.g., [7]). Twenty years ago Mabuchi [15] has proposed maximum likelihood methods to estimate Hamiltonian parameters. Later on, Gambetta and Wiseman [11] have given a first formu- lation of particle filtering techniques for classical pa- rameter estimation in open quantum systems. This for- mulation has been analyzed in [8] via an embedding in the standard quantum filtering formalism. Recently Ne- gretti and Mølmer [16] have exploited this embedding

∗This work has been partly funded by the Idex PSL * under the grant ANR-10-IDEX-0001-02 PSL *, by the Emergences program of Ville de Paris under the grant Qumotel and by the Projet Blanc ANR- 2011-BS01-017-01 EMAQS.

†Centre Automatique et Syst`emes, Mines-ParisTech, PSL Re- search University. 60, bd Saint-Michel 75006 Paris.

‡Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure-PSL Re- search University, CNRS, Universit´e Pierre et Marie Curie-Sorbonne Universit´es, Universit´e Paris Diderot-Sorbonne Paris Cit´e, 24 rue Lhomond, 75231 Paris Cedex 05, France

to derive the general equations of a particle quantum fil- ter for systems governed by stochastic master equations driven by Wiener processes (diffusive case). In these contributions, realistic simulations illustrate the interest of such filters for the estimation of continuous param- eters. In [14], similar filters are used for purely dis- crete parameters in order to discriminate between dif- ferent topologies of quantum networks. The Bayesian parameter estimation used in the measurement-based feedback experiment reported in [4] is in fact a spe- cial case of particle quantum filtering when the quantum states remain diagonal in the energy-level basis, reduce to populations and classical probabilities.

The contribution of this paper is twofold: with the- orem 2, we show that particle quantum filters are al- ways stable processes; with lemma 2, we propose and justify a new positivity preserving formulation in the diffusive case. This formulation is shown to provide an efficient algorithm for precisely estimating the detection efficiency from experimental heterodyne measurements of the fluorescence field that is emitted by a supercon- ducting qubit [5]. The statistics of the measurement out- comes generated by this system cannot be described by classical probabilities since the density operators at var- ious times do not commute. As far as we know, this is the first time that a particle quantum filter is applied to an experiment [6] whose measurement statistics are ruled by non-commutative quantum probabilities.

Section 2 is devoted to the discrete-time formula- tion. The specific structure of Markov models describ- ing open-quantum systems is presented. Then particle quantum filters are detailed and shown to be always sta- ble (theorem 2). Finally, the link with MaxLike ap- proach and the case of multiple measurement records are addressed. In section 3, a positivity preserving for- mulation of particle quantum filters is proposed for dif- fusive systems. The mathematical justifications of this formulation is given in lemma 2. In section 4, the nu- merical algorithm underlying lemma 2 is applied on ex- perimental data from which the detection efficiency is estimated and compared to an existing calibration pro-

tocol.

2. Discrete-time formulation

In the sequel, H is the finite-dimensional Hilbert space of the system and expectation values are denoted by the symbolE(.). In this section, time is indexed by the integer k=0,1, . . .The measurement outcome at k is denoted by y_k. It corresponds to a classical output sig- nal. We limit ourselves to the case where each y_k can take a finite set of values y_k∈ {1, . . . ,m}, m being a pos- itive integer (for continuous values of y, see section 3).

We denote byρ_kthe density operator at time-step k (an Hermitian operator onHsuch that Tr ρ_k=1,ρ_k≥0).

It corresponds to the conditional quantum state at time k knowing the initial conditionρ0and the past outcomes y1, . . . ,yk. According to the law of quantum mechan- ics,ρkis related toρk−1via the following Markov pro- cess (see, e.g., [22]) corresponding to a Davies instru- ment [9] in a discrete context:

ρ_k= K_yk(ρ_k−1) Tr

K_yk(ρ_k−1) (1) where the super-operator ρ7→K_y(ρ) depends on y, is linear and completely positive. It admits the follow- ing Kraus representation K_y(ρ)=P

µM_µ^yρ(M_µ^y)^†where the operators on H, (M_µ^y), satisfyP

µ,y(M^y_µ)^†M^y_µ=I_H with I_H the identity operator. Moreover the probability P y_k=y|ρ₀,y₁, . . . ,y_k−1 to detect y_k knowing the past outcomes and the initial stateρ₀, depends only onρ_k−1 (Markov property) and is given by

P y_k=y

ρ_k−1

=Tr

K_y(ρ_k−1) . Notice that E

ρk

ρk−1

= K(ρk−1) where K(ρ) = P

yK_y(ρ)=P

µ,yM_µ^yρ(M^y_µ)^†is a Kraus map (a quantum channel) since it is not only completely positive but also trace preserving: Tr K(ρ)=Tr ρ. In the sequel, K_yis called a partial Kraus map since it is not trace preserving in general: Tr

Ky(ρ)

≤Tr ρ. See, e.g., [10, 2] for a de- tailed construction of such Kybased on positive opera- tor value measures (POVM) and left stochastic matrices modeling measurement uncertainties and decoherence.

Now, we consider that the partial Kraus maps (K_y)_y=1,...,mcan depend on time k, (K_y,k), and on some physical parameters, grouped in the scalar or vectorial time-invariant p, (K_y,k^p ), whose exact value p may not be known with a sufficient precision, and whose esti- mation is the subject of this paper. Here, we consider the case where the only reliable resource of information

is some independent series of measurement outcomes, (y_k)_k=1,...,T, associated to a quantum trajectory of dura- tion T . Starting from the exact quantum state ρ₀ and the exact parameter value p, the exact quantum state trajectory (ρ_k)_k=1,...,T is given by the following Markov process:

ρ_k= K_y^p

k,k(ρ_k−1) Tr

K_y^p

k,k(ρ_k−1)

(2)

with the following probability of outcome y_k knowing ρ_k−1and p:

P yk=y

ρ_k−1,p

=Tr

K_y,k^p (ρ_k−1)

.

2.2. Particle quantum filters

The parameter estimation method described in [11, 8, 16] for continuous-time quantum trajectories admits the following discrete-time formulation. When the ex- act parameter value p and the initial state ρ₀ are un- known, one can still resort to the approximate filter cor- responding to its a priori estimate value p, with partial Kraus maps K_y^p

k,k, an initial guess forρ0and following statesρ_k^p satisfyingρ_k^p=

K^p

yk,k(ρ_k^p

−1) Tr

K^p

yk,k(ρ_k^p

−1)

. Here, the mea- surement outcomes (y_k)_k=1,...,T correspond to the hidden state Markov chain defined in (2) and involving the ac- tual value p of the parameter.

Assume that the initial information of the true pa- rameter value p is that it can take only two different values a or b. This initial uncertainty on the value of p can be taken into account by using an extended den- sity operator, denotedξ=diag(ξ^a, ξ^b), block diagonal, where the first blockξ^acorresponds to p=a, and the second blockξ^bto p=b. The evolution of each block is then handled with the corresponding partial Kraus maps (K^a_y,k) and (K^b_y,k) forming extended partial Kraus maps Ξy,k=diag

K^a_y,k,K^b_y,k

between block diagonal density operators on the Hilbert spaceH × H:

Ξy,k: ξ7→diag(K^a_y,k(ξ^a),K^b_y,k(ξ^b)). (3) The associated extended quantum filter reads:

ξ_k= Ξy_k,k(ξ_k−1) Tr

Ξy_k,k(ξ_k−1). (4) For p∈ {a,b}, the probability that p=p at step k knowing the initial quantum state ρ₀ and initial pa- rameter probability (π^a₀, π^b₀) reads π_k^p =Tr

ξ_k^p . In- deed, π^a_k+π^b_k =1 since Tr ξ =Tr ξ^a+Tr

ξ^b

=1, andξ0=diag(π^a₀ρ₀, π^b₀ρ₀). If the initial information on

the parameter value is only its belonging to{a,b}, then π^a₀=π^b₀=1/2.

Instead of usingξ=diag(ξ^a, ξ^b) itself, we decom- pose its terms into products of probabilitiesπ^pand den- sity operatorsρ^p=ξ^p/π^p. Then Eq. (4) reads























ρ_k^p= ^K

p yk,k(ρ_k^p

−1) Tr

K^p

yk,k(ρ_k^p

−1)

π_k^p=

Tr

K^p

yk,k(ρ_k^p

−1)

π_k^p

−1

Pp′∈{a,b}Tr

K^p′

yk,k(ρ_k^p′

−1)

π_k^p′

−1

(5)

for p∈ {a,b}. In the sequel, we will identify the filter stateξwith (ρ^a, ρ^b, π^a, π^b).

We have the following stability result based on [19, 21] and relying on the fidelity F(ρ, ρ^′)∈[0,1] between two density operatorsρandρ^′defined here as the square of the usual fidelity function used in quantum informa- tion [17]:

F(ρ, ρ^′)=Tr²

q√ρρ^′√ρ

! .

Theorem 1. Take an arbitrary initial quantum state ρ₀ and a parameter value p. Consider the quantum Markov process (2) producing the measurement record yk, k≥0. Assume that the constant parameter p can only take two different values, a and b. Consider the particle (quantum) filter (5) initialized withρ^a₀=ρ^b₀=ρ0

(ρ₀ any density operator) and (π^a₀, π^b₀)∈ [0,1]² with π^a₀+π^b₀ =1. Then F(ρ, ρ^p) and π^pF(ρ, ρ^p) are sub- martingales of the Markov process (2) and (5) of state (ρ, ρ^a, ρ^b, π^a, π^b):

Whenρ₀=ρ₀, we haveρ^p≡ρ, F(ρ, ρ^p)=1. Thus π^pis a sub-martingale

E

π_k^p

ρ_k−1, ξ_k−1

≥π_k^p

−1

This means that, in practice, the component ofπassoci- ated to the true value of the parameter tends to increase.

Proof. The fact that F(ρ, ρ^p) is a sub-martingale is a direct consequence of [21, theorem IV.1]: (ρ, ρ^p) is the state of the following quantum Markov chain

ρ_k= K_y^p

k,k(ρ_k−1) Tr

K_y^p

k,k(ρ_k−1)

, ρ_k^p= K_y^p

k,k(ρ_k^p

−1) Tr

K_y^p

k,k(ρ_k^p

−1)

with initial state (ρ₀, ρ₀) and measurement outcome y_k whose probabilityP

yk=y ρ_k−1

=Tr

K_y,k^p (ρ_k−1)

de- pends only onρ_k−1.

For instance, assume that p =a. Denote by ξ the state of the quantum filter (4) initialized with ξ₀=

diag(ρ₀,0). Thenξ≡(ρ,0) and thus (ξ, ξ) is solution of the extended Markov chain

ξ_k= Ξy_k,k(ξ_k−1) Tr

Ξ^yk,k(ξ_k−1), ξ_k= Ξy_k,k(ξk−1) Tr

Ξ^yk,k(ξk−1) with measurement outcome yk of probability P

y_k=y ξ_k−1

= Tr

Ξy,k(ξ_k−1)

depending only onξ_k−1. Thus according to [21, theorem IV.1], F(ξ, ξ) is a sub-martingale. Due to the block structure of ξ = diag(ρ,0) and ξ = diag(π^aρ^a, π^bρ^b), we have

F(ξ, ξ)=π^aF(ρ, ρ^a).

Extension of theorem 1 to an arbitrary number r of parameter values is given below, the proof being very similar and not detailed here.

Theorem 2. Take an arbitrary initial quantum stateρ₀ and parameter value p. Consider the quantum Markov process (2) producing the measurement record y_k, k≥0.

Assume that the parameter p belongs to a set of r differ- ent values (p_l)_l=1,...,r. Take, for l=1, . . . ,r, the particle quantum filter























ρ_k^pl = ^K

pl yk,k(ρ_k−1^pl ) Tr

K^pl

yk,k(ρ^pl

k−1)

π_k^pl= ^Tr

K^pl

yk,k(ρ_k^pl

−1)

π_k^pl

−1

Pr j=1Tr

K^{p j}

yk,k(ρ_k^{p j}

−1)

π_k^{p j}

−1

initialized withρ^p₀^l =ρ₀(ρ₀any density operator) and (π₀^p1, . . . , π₀^pr)∈[0,1]^rwithP

jπ₀^pj=1.

Then F(ρ, ρ^p) andπ^pF(ρ, ρ^p) are sub-martingales of the Markov process driven by (2) and of state (ρ, ρ^p1, . . . , ρ^pr, π^p1, . . . , π^pr):

Extension to a continuum of values for p of such particle quantum filters and of the above stability result can be done without major difficulties.

2.3. Connexion with MaxLike methods

Assume that the initial density operator is well known: ρ₀=ρ₀. It is possible to choose as an es- timation of p, among a or b, the value p that max- imises the probability π_k^p after a certain amount of time k. This method is actually a maximum-likelihood based technique. The multiplicative increment at time k for π^a_k is Tr

K^a_y

k,k(ρ^a_k

−1)

, which is equal to P

y_k

ρ₀,y₁, . . . ,y_k−1,p=a

. From this observation, we deduce that

π^a_k= π^a₀ C_k×

k

Y

l=1

P

y_l

ρ₀,y₁, . . . ,y_l−1,p=a

,

where C_kis a normalization factor to ensureπ^a_k+π^b_k=1.

Remarking that the probability of the measurement out- comes (y_l)_l≤kis the probability of the measurement out- comes (y_l)_l≤k−1times the probability of y_kconditionally to all prior measurements, one gets

π^a_k= π^a₀ C_k×P

y₁, . . . ,y_k

ρ₀,p=a

,

and similarly

π^b_k= π^b₀ C_k×P

y₁, . . . ,y_k

ρ₀,p=b

.

Choosing as an estimate the value a or b whose asso- ciated component ofπtends towards 1 thus amounts to choosing the parameter value that maximises the prob- ability of the measurement outcomes (y₁, . . . ,y_T).

2.4. Multiple quantum trajectories

Such particle quantum filtering techniques ex- tend without difficulties to N records (indexed by n∈ {1, . . .N}) of measurement outcomes, (y⁽ⁿ⁾_k )_k=1,...,Tn with possibly different lengths T_nand initial conditionsρ⁽ⁿ⁾₀ . This extension consists in a concatenation of the N records into a single record (¯y_k)_k=1,...,T with T=PN

n=1T_n and

(¯y_k)_k=1,...,T =

y⁽¹⁾₁ , . . . ,y⁽¹⁾_T

1,y⁽²⁾₁ , . . . ,y⁽²⁾_T

2, . . . ,y^(N)₁ , . . . ,y^(N)_T

N

This record can be associated to a single quantum tra- jectory of length T of form (2). First initialize atρ⁽¹⁾₀ . Then for each k equal to T₁+. . .+T_n−1,ρ_k+1is reset to ρ⁽ⁿ⁾₀ . This can be done by applying a reset Kraus map K^ρ(n)0 after the computation ofρ_k+1relying on outcome y⁽ⁿ_T⁻¹⁾

n−1 and before using the outcome y⁽ⁿ⁾₁ . For any den- sity operatorσ, it is simple to construct via its spec- tral decomposition, a Kraus map K^σsuch that, for all density operatorρ, K^σ(ρ)=σ. With this trick (¯yk) is associated to an effective single quantum trajectory of the form (2) where the partial Kraus maps K_y,k^p depend effectively on the time step k because of adding these reset Kraus maps.

For the particle quantum filter that is described in theorem 2 and associated to the record (¯y_k) , eachρ^(p_k^l)is reset in a similar way at each time step k=T₁+. . .+T_n−1 contrarily to the parameter probabilityπ^(p_k^l) that is not reset.

3. Continuous-time formulation

For a mathematical and precise description of such diffusive models, see [3]. We just recall here the stochastic master equation governing the time evolution of the density operator t7→ρt

dρt=

−i[H, ρt]+

m

X

ν=1

Dν(ρt)

dt

+

m

X

ν=1

√ην

Lνρt+ρtL^†_ν−Tr

Lνρt+ρtL^†_ν ρt

dW_t^ν (6)

where H is the Hamiltonian, an Hermitian operator on H(¯h=1 here) and where, for eachν∈ {1, . . . ,m},

• Dνis the Lindblad super-operator

Dν(ρ)=L_νρL^†_ν−¹₂(L^†_νL_νρ+ρL^†_νL_ν);

• L_ν is an operator on H, which is not necessarily Hermitian and which is associated to the measure- ment/decoherence channelν;

• ην ∈[0,1] is the detection efficiency (ην=0 for decoherence channel andη_ν>0 for measurement channel) ;

• W_t^ν is a Wiener process (independent of the other Wiener processes W_t^µ,ν) describing the quantum fluctuations of the continuous output signal t7→y^ν_t. It is related toρ_tby

dy^ν_t = √η_νTr

L_νρ_t+ρ_tL^†_ν

dt+dW_t^ν. (7) 3.2. Partial Kraus map formulation

We introduce here another formulation of (6) that mimics the discrete-time formulation (2). This formu- lation is inspired of subsection 4.3.3 of [12], subsec- tion entitled ”Physical interpretation of the master equa- tion”. In (6), dρ_t stands for ρ_t+dt−ρ_t. It can thus be written as

ρ_t+dt=ρt+

−i[H, ρt]+

m

X

ν=1

Dν(ρt)

dt

+

m

X

ν=1

√ην

Lνρt+ρtL^†_ν−Tr

Lνρt+ρtL^†_ν ρt

dW_t^ν i.e.,ρt+dtis an algebraic expression involvingρt, dt and dW_t^ν. With this form, it is not obvious thatρt+dtremains

a density operator ifρ_t is a density operator. The fol- lowing lemma provides another formulation based on It¯o calculus showing directly thatρ_t+dt remains a den- sity operator. In [20], similar formulations are proposed without the mathematical justifications given below and are tested in realistic simulations of measurement-based feedback scheme.

Lemma 1. Consider the stochastic differential equa- tion (6) with an initial condition ρ0, which is a non- negative Hermitian operator of trace one. Then it also reads:

ρ_t+dt= K_dyt,dt(ρ_t) Tr

K_dyt,dt(ρ_t),

where dy_tstands for (dy¹_t, . . . ,dy^m_t ), and where K_∆y,∆tis a partial Kraus map depending on∆y∈R^mand∆t>0 given by

K_∆y,∆t(ρ)=M_∆y,∆tρM^†_∆y,∆t+

m

X

ν=1

(1−η_ν)∆t L_νρL^†_ν and M_∆y,∆tis the following operator onH

M_∆y,∆t=I_H−









 iH+

m

X

ν=1

L^†_νL_ν/2











∆t+

m

X

ν=1

√η_ν∆y^νL_ν

Proof. Assume that m=1. Then, dρ_t=

−i[H, ρ_t]+Lρ_tL^†−¹2(L^†Lρ_t+ρ_tL^†L)

dt +√η

Lρ_t+ρ_tL^†−Tr

Lρ_t+ρ_tL^† ρ_t

dW_t. (8) Using It¯o rules, dy²_t =dt. Hence, we have

K_dyt,dt(ρ_t)=ρ_t+√η(Lρ_t+ρ_tL^†) dy_t +

−i[H, ρ_t]+Lρ_tL^†−¹2(L^†Lρ_t+ρ_tL^†L) dt.

Thus Tr

K_dyt,dt(ρ_t)

=1+√ηTr

Lρ_t+ρ_tL^† dy_tand 1

Tr

K_dyt,dt(ρ_t)=1−√ηTr

Lρt+ρtL^† dy_t +ηTr²

Lρ_t+ρ_tL^† dt.

We get K_dyt,dt(ρ_t) Tr

K_dyt,dt(ρ_t)−ρ_t

=√η

Lρ_t+ρ_tL^†−Tr

Lρ_t+ρ_tL^† ρ_t

dy_t +

−i[H, ρt]+LρtL^†−¹2(L^†Lρt+ρtL^†L) dt

−ηTr

Lρ_t+ρ_tL^†

Lρ_t+ρ_tL^†−Tr

Lρ_t+ρ_tL^† ρ_t

dt.

One recognizes (8) since dy_t−√ηTr

Lρ_t+ρ_tL^† dt= dW_t. For m>1, the computations are similar and not

detailed here.

3.3. Particle quantum filtering

Assume the system dynamics depends on a con- stant parameter p appearing either in the SME (6) and/or in the output maps (7). As in section 2, assume that p can take a finite number r of values p₁, . . . , p_r. Denote byρ_t^pthe quantum state associated to p:

dρ_t^p=L^p(ρ_t^p) dt+

m

X

ν=1

M^p(ρ^p_t) dW_t^ν (9) where the super-operators

L^p(ρ)=−i[H^p, ρ]

+ Xm

ν=1

L_ν^pρ(L_ν^p)^†−1

2((L_ν^p)^†L^p_νρ+ρ(L_ν^p)^†L_ν^p) and

M^p(ρ)= q

η_ν^p

L_ν^pρ+ρ(L_ν^p)^†−Tr

L_ν^pρ+ρ(L_ν^p)^† ρ

depend on p since the operators L_ν^pand the efficiencies η_ν^pcould depend on p. The m outputs that are associated to the parameter p then read:

dy^ν_t =C_ν^p(ρ_t^p)dt+dW_t^ν (10) forν=1, . . . ,m, and where:

C_ν^p(ρ)= q

η_ν^pTr

L_ν^pρ+ρ(L_ν^p)^† .

With these notations, the particle quantum filter in- troduced in [11] and further developed and analyzed in [8, 16] reads as follows. For each l∈ {1, . . . ,r}, ρ_t^pl is governed by the quantum filter:

dρ_t^pl=L^pl(ρ_t^pl) dt +

Xm

ν=1

M^pl(ρ_t^pl)

dy^ν_t−C_ν^pl(ρ_t^pl)dt , (11)

and the parameter probabilityπ_t^pl is governed by:

dπ_t^pl = π^p_t^l











m

X

ν=1

C_ν^pl(ρ_t^pl)−C^ν_t dy^ν_t−C^ν_tdt









 , (12)

where C^ν_t =Pr

j=1π_t^pjC_ν^pj(ρ_t^pj).

Here again, the lemma below provides another for- mulation of this particle quantum filter that mimics the discrete-time setting of theorem 2.

Lemma 2. For each l∈ {1, . . . ,r}, the particle quantum filter (11) and (12) can be formulated as follows:























ρ_t+dt^pl = ^K

pl dyt,dt(ρ_t^pl) Tr

K^pl

dyt,dt(ρ_t^pl)

π_t+dt^pl = ^Tr

K^pl

dyt,dt(ρ_t^pl)

π_t^pl Pr

j=1Tr

K^{p j}

dyt,dt(ρ_t^{p j})

π_t^{p j}

where dy_tstands for (dy¹_t, . . . ,dy^m_t) and where K_∆y,∆t^p is a partial Kraus map depending on p,∆y∈R^mand∆t>0 given by:

K^p

∆y,∆t(ρ)=M^p

∆y,∆tρ

M^p

∆y,∆t

_† +

Xm

ν=1

(1−η_ν^p)∆t L_ν^pρ(L_ν^p)^†,

and M^p

∆y,∆tis the following operator onH: M^p

∆y,∆t=I_H−









 iH^p+

m

X

ν=1

(L_ν^p)^†L_ν^p/2











∆t+

m

X

ν=1

q

η_ν^p∆y^νL_ν^p.

The proof is very similar to the proof of lemma 1.

It relies on simple but slightly tedious computations ex- ploiting It¯o calculus. Due to space limitation, this proof is not detailed here. This lemma, combined with the mathematical machineries exploited in [1], opens the way to an extension to the diffusive case of theorem 2.

4. An experimental validation

The estimation of the detection efficiency is con- ducted on a superconducting qubit whose fluorescence field is measured using a heterodyne detector [18, 13].

For the detailed physics of this experiment, see [5, 6].

The Hilbert spaceHisC². The system dynamics is de- scribed by a stochastic master equation of the form (6), with m=3:η₁=η₂=ηis the total efficiency of the het- erodyne measurement of the fluorescence signal;η₃=0 corresponds to an unmonitored dephasing channel:

L₁= q 1

2T₁

X−iY

2 , L₂=iL₁, L₃= q

1 2T_φZ where X, Y and Z are the usual Pauli matrices [17].

The time constants T₁=4.15 µs and T_φ=35 µs are determined independently using Rabi or Ramsey pro- tocols, which is not the case ofη. Using a calibration of the average resonance fluorescence signal, the mea- sured vacuum noise fluctuations provide a first estima- tion ofη=0.26±0.02.

To get a more precise estimation ofη, we have mea- sured N=3×10⁶quantum trajectories of 10µs, starting from the same known initial stateρ₀=^IH₂^+X. The sam- pling time ∆t is equal to 0.20µs. For each trajectory,

0 500 1000 1500 2000

0 0.2 0.4 0.6 0.8 1

Number of Trajectories π t(i)

η=0.1 η=0.26 η=0.4

Figure 1: First estimation, with pattern valuesη1=0.10, the parameter valueη2=0.26 close toη, andη3=0.40.

Only the first 2000 trajectories are needed to selectη≈ 0.26 and discard 0.10 and 0.40.

the measurement sample at time t_k=k∆t, k∈ {1, . . . ,50}, corresponds to the two quadratures of the fluorescence field∆y¹_k=y¹_k∆t−y¹_(k

−1)∆tand∆y²_k=y²_k∆t−y²_(k

−1)∆t. From lemma 2, we derive a simple recursive algorithm where (dy_t) and dt are replaced by (∆y_k) and∆t. Moreover, as explained in subsection 2.4, the 3×10⁶quantum trajec- tories are concatenated into a single one.

The estimation is done by taking some pattern val- uesη₁,η₂, ...,η_r, assuming that the real valueηis suffi- ciently close to one of them. We begin with a first esti- mation that keeps a big interval between each possible valueη_iofη, in order to validate our estimation scheme.

We then sharpen this estimation by reducing the inter- vals between each valueη_i, until arriving to a level of accuracy after which no distinct discrimination can be performed. The results are given at figures 1, 2 and 3.

They give the following refinement of the initial cali- bration:η=0.2425±0.005. On each of the figures, the X-axis represents the number of trajectories after which we look at the parameter probabilitiesπ^η_kⁱand the Y-axis displays these probabilities.

5. Conclusion

We have shown that particle quantum filtering is always a stable process. We have proposed an origi- nal positivity preserving formulation for systems gov- erned by diffusive stochastic master equation. A first validation on experimental data confirms the interest of the resulting parameter algorithm. This positivity pre- serving algorithm appears to be robust enough to cope with sampling time of more than 2% of the characteris- tic time attached to the measurement. The convergence

0 2 4 6 8 10 x 10⁴ 0

0.2 0.4 0.6 0.8 1

Number of Trajectories π t(i)

η=0.22 η=0.24 η=0.26 η=0.28

Figure 2: Second estimation, realized with more nar- row intervals between each pattern values. We notice thatηis actually closer to 0.24 than 0.26, the calibrated value, and that the number of trajectories required for the discrimination has drastically increased to 1×10⁵.

0 0.5 1 1.5 2 2.5 3

x 10⁶ 0

0.2 0.4 0.6 0.8 1

Number of Trajectories π t(i)

η=0.23 η=0.235 η=0.24 η=0.245 η=0.25

Figure 3: Last estimation, with very narrow intervals.

We use all the trajectories available, i.e. 3×10⁶ tra- jectories. Filter does not converge to a distinct choice between 0.240 and 0.245.

characterization of such estimation scheme remains to be done despite the fact they are always stable.

Acknowledgment

The authors thank Michel Brune, Igor Dotsenko and Jean-Michel Raimond for useful discussions on quantum filtering and parameter estimation in the discrete-time case.

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